Grade 12 · Mathematics
Differential calculus
Use derivatives to find gradients, tangent equations and stationary points of polynomial functions.
Differentiate each term
For a polynomial term axⁿ, the derivative is anxⁿ⁻¹. A constant differentiates to zero. In this guide the power rule is used for polynomial functions.
Interpret the derivative
At x = a, f′(a) is the tangent gradient. Solve f′(x) = 0 for stationary points, then substitute each x value into f(x) for its y coordinate.
Formula sheet
- d(axⁿ)/dx = anxⁿ⁻¹
- Tangent: y - f(a) = f′(a)(x - a)
Worked example
Find the tangent to f(x) = x² + 3x at x = 1.
- Differentiate: f′(x) = 2x + 3.
- Gradient at x = 1: f′(1) = 5.
- Point on the graph: f(1) = 4, so (1; 4).
- Tangent: y - 4 = 5(x - 1), hence y = 5x - 1.
Try it yourself
1. Differentiate f(x) = 3x³ - 4x + 7. Check answer
f′(x) = 9x² - 4.
2. Find the stationary point of f(x) = x² - 6x + 2. Check answer
f′(x) = 2x - 6 = 0 gives x = 3; f(3) = -7. The point is (3; -7).
Common mistake: A derivative gives the gradient, not the y coordinate. Use the original function to find the point.